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<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 3.2 Final//EN"> <html> <head> <link rel="stylesheet" href="/styles.css"> <meta name="format-detection" content="telephone=no"> <meta http-equiv="content-type" content="text/html; charset=utf-8"> <meta name=viewport content="width=device-width, initial-scale=1"> <meta name="keywords" content="OEIS,integer sequences,Sloane" /> <title>A002865 - OEIS</title> <link rel="search" type="application/opensearchdescription+xml" title="OEIS" href="/oeis.xml"> <script> var myURL = "\/A002865" function redir() { var host = document.location.hostname; if(host != "oeis.org" && host != "127.0.0.1" && !/^([0-9.]+)$/.test(host) && host != "localhost" && host != "localhost.localdomain") { document.location = "https"+":"+"//"+"oeis"+".org/" + myURL; } } function sf() { if(document.location.pathname == "/" && document.f) document.f.q.focus(); } </script> </head> <body bgcolor=#ffffff onload="redir();sf()"> <div class=loginbar> <div class=login> <a href="/login?redirect=%2fA002865">login</a> </div> </div> <div class=center><div class=top> <center> <div class=donors> The OEIS is supported by <a href="http://oeisf.org/#DONATE">the many generous donors to the OEIS Foundation</a>. </div> <div class=banner> <a href="/"><img class=banner border="0" width="600" src="/banner2021.jpg" alt="A002865 - OEIS"></a> </div> </center> </div></div> <div class=center><div class=pagebody> <div class=searchbarcenter> <form name=f action="/search" method="GET"> <div class=searchbargreet> <div class=searchbar> <div class=searchq> <input class=searchbox maxLength=1024 name=q value="" title="Search Query"> </div> <div class=searchsubmit> <input type=submit value="Search" name=go> </div> <div class=hints> <span class=hints><a href="/hints.html">Hints</a></span> </div> </div> <div class=searchgreet> (Greetings from <a href="/welcome">The On-Line Encyclopedia of Integer Sequences</a>!) </div> </div> </form> </div> <div class=sequence> <div class=space1></div> <div class=line></div> <div class=seqhead> <div class=seqnumname> <div class=seqnum> A002865 </div> <div class=seqname> Number of partitions of n that do not contain 1 as a part. <br><font size=-1>(Formerly M0309 N0113)</font> </div> </div> <div class=scorerefs> 408 </div> </div> <div> <div class=seqdatabox> <div class=seqdata>1, 0, 1, 1, 2, 2, 4, 4, 7, 8, 12, 14, 21, 24, 34, 41, 55, 66, 88, 105, 137, 165, 210, 253, 320, 383, 478, 574, 708, 847, 1039, 1238, 1507, 1794, 2167, 2573, 3094, 3660, 4378, 5170, 6153, 7245, 8591, 10087, 11914, 13959, 16424, 19196, 22519, 26252, 30701</div> <div class=seqdatalinks> (<a href="/A002865/list">list</a>; <a href="/A002865/graph">graph</a>; <a href="/search?q=A002865+-id:A002865">refs</a>; <a href="/A002865/listen">listen</a>; <a href="/history?seq=A002865">history</a>; <a href="/search?q=id:A002865&fmt=text">text</a>; <a href="/A002865/internal">internal format</a>) </div> </div> </div> <div class=entry> <div class=section> <div class=sectname>OFFSET</div> <div class=sectbody> <div class=sectline>0,5</div> </div> </div> <div class=section> <div class=sectname>COMMENTS</div> <div class=sectbody> <div class=sectline>Also the number of partitions of n-1, n >= 2, such that the least part occurs exactly once. See <a href="/A096373" title="Number of partitions of n such that the least part occurs exactly twice.">A096373</a>, <a href="/A097091" title="Number of partitions of n such that the least part occurs exactly three times.">A097091</a>, <a href="/A097092" title="Number of partitions of n such that the least part occurs exactly four times.">A097092</a>, <a href="/A097093" title="Number of partitions of n such that the least part occurs exactly five times.">A097093</a>. - <a href="/wiki/User:Robert_G._Wilson_v">Robert G. Wilson v</a>, Jul 24 2004 [Corrected by <a href="/wiki/User:Wolfdieter_Lang">Wolfdieter Lang</a>, Feb 18 2009]</div> <div class=sectline>Number of partitions of n+1 where the number of parts is itself a part. Take a partition of n (with k parts) which does not contain 1, remove 1 from each part and add a new part of size k+1. - <a href="/wiki/User:Franklin_T._Adams-Watters">Franklin T. Adams-Watters</a>, May 01 2006</div> <div class=sectline>Number of partitions where the largest part occurs at least twice. - <a href="/wiki/User:Joerg_Arndt">Joerg Arndt</a>, Apr 17 2011</div> <div class=sectline>Row sums of triangle <a href="/A147768" title="Triangle read by rows: A000012^(-2) * A027293 as infinite lower triangular matrices.">A147768</a>. - <a href="/wiki/User:Gary_W._Adamson">Gary W. Adamson</a>, Nov 11 2008</div> <div class=sectline>From Lewis Mammel (l_mammel(AT)att.net), Oct 06 2009: (Start)</div> <div class=sectline>a(n) is the number of sets of n disjoint pairs of 2n things, called a pairing, disjoint with a given pairing (<a href="/A053871" title="a(0)=1; a(1)=0; a(n) = 2*(n-1)*(a(n-1) + a(n-2)).">A053871</a>), that are unique under permutations preserving the given pairing.</div> <div class=sectline>Can be seen immediately from a graphical representation which must decompose into even numbered cycles of 4 or more things, as connected by pairs alternating between the pairings. Each thing is in a single cycle, so this is a partition of 2n into even parts greater than 2, equivalent to a partition of n into parts greater than 1. (End)</div> <div class=sectline>Convolution product (1, 1, 2, 2, 4, 4, ...) * (1, 2, 3, ...) = <a href="/A058682" title="a(n) = p(0) + p(1) + ... + p(n) - n - 1, where p = partition numbers, A000041.">A058682</a> starting (1, 3, 7, 13, 23, 37, ...); with row sums of triangle <a href="/A171239" title="Triangle read by rows extracted from convolution product (1,2,3,...) * A002865: (1,1,2,2,4,4,7,8,...)">A171239</a> = <a href="/A058682" title="a(n) = p(0) + p(1) + ... + p(n) - n - 1, where p = partition numbers, A000041.">A058682</a>. - <a href="/wiki/User:Gary_W._Adamson">Gary W. Adamson</a>, Dec 05 2009</div> <div class=sectline>Also the number of 2-regular multigraphs with loops forbidden. - <a href="/wiki/User:Jason_Kimberley">Jason Kimberley</a>, Jan 05 2011</div> <div class=sectline>Number of appearances of the multiplicity n, n-1, ..., n-k in all partitions of n, for k < n/2. (Only populated by multiplicities of large numbers of 1's.) - William Keith, Nov 20 2011</div> <div class=sectline>Also the number of equivalence classes of n X n binary matrices with exactly 2 1's in each row and column, up to permutations of rows and columns (cf. <a href="/A133687" title="Triangle with number of equivalence classes of n X n matrices over {0,1} with rows and columns summing to k (0<=k<=n), where...">A133687</a>). - <a href="/wiki/User:N._J._A._Sloane">N. J. A. Sloane</a>, Sep 16 2013</div> <div class=sectline>The q-Catalan numbers ((1-q)/(1-q^(n+1)))[2n,n]_q, where [2n,n]_q are the central q-binomial coefficients, match this sequence in their initial segment of length n. - <a href="/wiki/User:William_J._Keith">William J. Keith</a>, Nov 14 2013</div> <div class=sectline>Starting at a(2) this sequence gives the number of vertices on a nim tree created in the game of edge removal for a path P_{n} where n is the number of vertices on the path. This is the number of nonisomorphic graphs that can result from the path when the game of edge removal is played. - <a href="/wiki/User:Lyndsey_Wong">Lyndsey Wong</a>, Jul 09 2016</div> <div class=sectline>The number of different ways to climb a staircase taking at least two stairs at a time. - <a href="/wiki/User:Mohammad_K._Azarian">Mohammad K. Azarian</a>, Nov 20 2016</div> <div class=sectline>Let 1,0,1,1,1,... (offset 0) count unlabeled, connected, loopless 1-regular digraphs. This here is the Euler transform of that sequence, counting unlabeled loopless 1-regular digraphs. <a href="/A145574" title="Array a(n,m) for number of partitions of n>=2 with m parts having no part 1. Hence m=1..floor(n/2).">A145574</a> is the associated multiset transformation. <a href="/A000166" title="Subfactorial or rencontres numbers, or derangements: number of permutations of n elements with no fixed points.">A000166</a> are the labeled loopless 1-regular digraphs. - <a href="/wiki/User:R._J._Mathar">R. J. Mathar</a>, Mar 25 2019</div> <div class=sectline>For n > 1, also the number of partitions with no part greater than the number of ones. - <a href="/wiki/User:George_Beck">George Beck</a>, May 09 2019 [See <a href="/A187219" title="Number of partitions of n that do not contain parts less than the smallest part of the partitions of n-1.">A187219</a> which is the correct sequence for this interpretation for n >= 1. - <a href="/wiki/User:Spencer_Miller">Spencer Miller</a>, Jan 30 2023]</div> <div class=sectline>From <a href="/wiki/User:Gus_Wiseman">Gus Wiseman</a>, May 19 2019: (Start)</div> <div class=sectline>Conjecture: Also the number of integer partitions of n - 1 that have a consecutive subsequence summing to each positive integer from 1 to n - 1. For example, (32211) is such a partition because we have consecutive subsequences:</div> <div class=sectline> 1: (1)</div> <div class=sectline> 2: (2)</div> <div class=sectline> 3: (3) or (21)</div> <div class=sectline> 4: (22) or (211)</div> <div class=sectline> 5: (32) or (221)</div> <div class=sectline> 6: (2211)</div> <div class=sectline> 7: (322)</div> <div class=sectline> 8: (3221)</div> <div class=sectline> 9: (32211)</div> <div class=sectline>(End)</div> <div class=sectline>There is a sufficient and necessary condition to characterize the partitions defined by Gus Wiseman. It is that the largest part must be less than or equal to the number of ones plus one. Hence, the number of partitions of n with no part greater than the number of ones is the same as the number of partitions of n-1 that have a consecutive subsequence summing to each integer from 1 to n-1. Gus Wiseman's conjecture can be proved bijectively. - <a href="/wiki/User:Andrew_Yezhou_Wang">Andrew Yezhou Wang</a>, Dec 14 2019</div> <div class=sectline>From <a href="/wiki/User:Peter_Bala">Peter Bala</a>, Dec 01 2024: (Start)</div> <div class=sectline>Let P(2, n) denote the set of partitions of n into parts k > 1. Then <a href="/A000041" title="a(n) is the number of partitions of n (the partition numbers).">A000041</a>(n) = - Sum_{parts k in all partitions in P(2, n+2)} mu(k). For example, with n = 5, there are 4 partitions of n + 2 = 7 into parts greater than 1, namely, 7, 5 + 2, 4 + 3, 3 + 2 + 2, and mu(7) + (mu(5) + mu(2)) + (mu(4 ) + mu(3)) + (mu(3) + mu(2) + mu(2)) = -7 = - <a href="/A000041" title="a(n) is the number of partitions of n (the partition numbers).">A000041</a>(5). (End)</div> </div> </div> <div class=section> <div class=sectname>REFERENCES</div> <div class=sectbody> <div class=sectline>M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 836.</div> <div class=sectline>L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 115, p*(n).</div> <div class=sectline>H. P. Robinson, Letter to N. J. A. Sloane, Jan 04 1974.</div> <div class=sectline>N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).</div> <div class=sectline>N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).</div> <div class=sectline>P. G. Tait, Scientific Papers, Cambridge Univ. Press, Vol. 1, 1898, Vol. 2, 1900, see Vol. 1, p. 334.</div> </div> </div> <div class=section> <div class=sectname>LINKS</div> <div class=sectbody> <div class=sectline>Andrew van den Hoeven, <a href="/A002865/b002865.txt">Table of n, a(n) for n = 0..10000</a> (first 1001 terms from T. D. Noe)</div> <div class=sectline>M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].</div> <div class=sectline>A. P. Akande et al., <a href="https://arxiv.org/abs/2112.03264">Computational study of non-unitary partitions</a>, arXiv:2112.03264 [math.CO], 2021.</div> <div class=sectline>Colin Albert, Olivia Beckwith, Irfan Demetoglu, Robert Dicks, John H. Smith, and Jasmine Wang, <a href="https://arxiv.org/abs/2203.08987">Integer partitions with large Dyson rank</a>, arXiv:2203.08987 [math.NT], 2022.</div> <div class=sectline>Max A. Alekseyev and Allan Bickle, <a href="https://allanbickle.wordpress.com/wp-content/uploads/2016/05/forbidsubfinal.pdf">Forbidden Subgraphs of Single Graphs</a>, (2024). See p. 6.</div> <div class=sectline>Kevin Beanland and Hung Viet Chu, <a href="https://arxiv.org/abs/2311.01926">On Schreier-type Sets, Partitions, and Compositions</a>, arXiv:2311.01926 [math.CO], 2023.</div> <div class=sectline>G. Dahl and T. A. Haufmann, <a href="https://www.researchgate.net/publication/303684695_Zero-one_completely_positive_matrices_and_the_AR_S_classes">Zero-one completely positive matrices and the A(R,S) classes</a>, Preprint, 2016.</div> <div class=sectline>R. P. Gallant, G. Gunther, B. L. Hartnell, and D. F. Rall, <a href="https://www.researchgate.net/publication/268659207">A game of edge removal on graphs</a>, JCMCC, 57 (2006), 75 - 82.</div> <div class=sectline>Edray Herber Goins and Talitha M. Washington, <a href="https://arxiv.org/abs/0909.5459">On the generalized climbing stairs problem</a>, Ars Combin. 117 (2014), 183-190. MR3243840 (Reviewed), arXiv:0909.5459 [math.CO], 2009.</div> <div class=sectline>H. Gropp, <a href="http://dx.doi.org/10.1016/0012-365X(94)00372-P">On tactical configurations, regular bipartite graphs and (v,k,even)-designs</a>, Discr. Math., 155 (1996), 81-98.</div> <div class=sectline>R. K. Guy and N. J. A. Sloane, <a href="/A005180/a005180.pdf">Correspondence</a>, 1988.</div> <div class=sectline>Cristiano Husu, <a href="https://arxiv.org/abs/1804.09883">The butterfly sequence: the second difference sequence of the numbers of integer partitions with distinct parts, its pentagonal number structure, its combinatorial identities and the cyclotomic polynomials 1-x and 1+x+x^2</a>, arXiv:1804.09883 [math.NT], 2018.</div> <div class=sectline>INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=100">Encyclopedia of Combinatorial Structures 100</a></div> <div class=sectline>Wenwei Li, <a href="http://arxiv.org/abs/1612.05526">Approximation of the Partition Number After Hardy and Ramanujan: An Application of Data Fitting Method in Combinatorics</a>, arXiv preprint arXiv:1612.05526 [math.NT], 2016-2018.</div> <div class=sectline>Wenwei Li, <a href="https://arxiv.org/abs/1612.08186">On the Number of Conjugate Classes of Derangements</a>, arXiv:1612.08186 [math.CO], 2016.</div> <div class=sectline>J. L. Nicolas and A. S谩rk枚zy, <a href="http://www.numdam.org/item?id=JTNB_2000__12_1_227_0">On partitions without small parts</a>, Journal de th茅orie des nombres de Bordeaux, 12 no. 1 (2000), p. 227-254.</div> <div class=sectline>R. A. Proctor, <a href="https://arxiv.org/abs/math/0606404">Let's Expand Rota's Twelvefold Way for Counting Partitions!</a>, arXiv:math/0606404 [math.CO], 2006-2007.</div> <div class=sectline>H. P. Robinson, <a href="/A002065/a002065.pdf">Letter to N. J. A. Sloane, Jul 12 1971</a></div> <div class=sectline>H. P. Robinson, <a href="/A002865/a002865.pdf">Letter to N. J. A. Sloane, Dec 10 1973</a></div> <div class=sectline>H. P. Robinson, <a href="/A003105/a003105.pdf">Letter to N. J. A. Sloane, Jan 4 1974</a>.</div> <div class=sectline>Noah Rubin, Curtis Bright, Kevin K. H. Cheung, and Brett Stevens, <a href="https://arxiv.org/abs/2103.11018">Integer and Constraint Programming Revisited for Mutually Orthogonal Latin Squares</a>, arXiv:2103.11018 [cs.DM], 2021.</div> <div class=sectline>Miloslav Znojil, <a href="https://arxiv.org/abs/2010.15014">Non-Hermitian N-state degeneracies: unitary realizations via antisymmetric anharmonicities</a>, arXiv:2010.15014 [quant-ph], 2020.</div> <div class=sectline>Miloslav Znojil, <a href="https://arxiv.org/abs/2102.12272">Quantum phase transitions mediated by clustered non-Hermitian degeneracies</a>, arXiv:2102.12272 [quant-ph], 2021.</div> <div class=sectline>Miloslav Znojil, <a href="https://arxiv.org/abs/2108.07110">Bose-Einstein condensation processes with nontrivial geometric multiplicites realized via PT-symmetric and exactly solvable linear-Bose-Hubbard building blocks</a>, arXiv:2108.07110 [quant-ph], 2021.</div> <div class=sectline><a href="/index/Par#partN">Index entries for related partition-counting sequences</a></div> </div> </div> <div class=section> <div class=sectname>FORMULA</div> <div class=sectbody> <div class=sectline>G.f.: Product_{m>1} 1/(1-x^m).</div> <div class=sectline>a(0)=1, a(n) = p(n) - p(n-1), n >= 1, with the partition numbers p(n) := <a href="/A000041" title="a(n) is the number of partitions of n (the partition numbers).">A000041</a>(n).</div> <div class=sectline>a(n) = <a href="/A085811" title="Number of partitions of n including 3, but not 1.">A085811</a>(n+3). - <a href="/wiki/User:James_A._Sellers">James A. Sellers</a>, Dec 06 2005 [Corrected by <a href="/wiki/User:Gionata_Neri">Gionata Neri</a>, Jun 14 2015]</div> <div class=sectline>a(n) = <a href="/A116449" title="Number of partitions of n into an equal number of prime and composite parts.">A116449</a>(n) + <a href="/A116450" title="Number of partitions of n such that the numbers of prime and composite parts differ by at least 1.">A116450</a>(n). - <a href="/wiki/User:Reinhard_Zumkeller">Reinhard Zumkeller</a>, Feb 16 2006</div> <div class=sectline>a(n) = Sum_{k=2..floor((n+2)/2)} <a href="/A008284" title="Triangle of partition numbers: T(n,k) = number of partitions of n in which the greatest part is k, 1 <= k <= n. Also number ...">A008284</a>(n-k+1,k-1) for n > 0. - <a href="/wiki/User:Reinhard_Zumkeller">Reinhard Zumkeller</a>, Nov 04 2007</div> <div class=sectline>G.f.: 1 + Sum_{n>=2} x^n / Product_{k>=n} (1 - x^k). - <a href="/wiki/User:Joerg_Arndt">Joerg Arndt</a>, Apr 13 2011</div> <div class=sectline>G.f.: Sum_{n>=0} x^(2*n) / Product_{k=1..n} (1 - x^k). - <a href="/wiki/User:Joerg_Arndt">Joerg Arndt</a>, Apr 17 2011</div> <div class=sectline>a(n) = <a href="/A090824" title="Triangle read by rows: T(n,k) = number of partitions of binomial(n,k) into parts greater than k and not greater than n, 0<=k<=n.">A090824</a>(n,1) for n > 0. - <a href="/wiki/User:Reinhard_Zumkeller">Reinhard Zumkeller</a>, Oct 10 2012</div> <div class=sectline>a(n) ~ Pi * exp(sqrt(2*n/3)*Pi) / (12*sqrt(2)*n^(3/2)) * (1 - (3*sqrt(3/2)/Pi + 13*Pi/(24*sqrt(6)))/sqrt(n) + (217*Pi^2/6912 + 9/(2*Pi^2) + 13/8)/n). - <a href="/wiki/User:Vaclav_Kotesovec">Vaclav Kotesovec</a>, Feb 26 2015, extended Nov 04 2016</div> <div class=sectline>G.f.: exp(Sum_{k>=1} (sigma_1(k) - 1)*x^k/k). - <a href="/wiki/User:Ilya_Gutkovskiy">Ilya Gutkovskiy</a>, Aug 21 2018</div> <div class=sectline>a(0) = 1, a(n) = <a href="/A232697" title="Number of partitions of 2n into parts such that the largest multiplicity equals n.">A232697</a>(n) - 1. - <a href="/wiki/User:George_Beck">George Beck</a>, May 09 2019</div> <div class=sectline>From <a href="/wiki/User:Peter_Bala">Peter Bala</a>, Feb 19 2021: (Start)</div> <div class=sectline>G.f.: A(q) = Sum_{n >= 0} q^(n^2)/( (1 - q)*Product_{k = 2..n} (1 - q^k)^2 ).</div> <div class=sectline>More generally, A(q) = Sum_{n >= 0} q^(n*(n+r))/( (1 - q) * Product_{k = 2..n} (1 - q^k)^2 * Product_{i = 1..r} (1 - q^(n+i)) ) for r = 0,1,2,.... (End)</div> <div class=sectline>G.f.: 1 + Sum_{n >= 1} x^(n+1)/Product_{k = 1..n-1} 1 - x^(k+2). - <a href="/wiki/User:Peter_Bala">Peter Bala</a>, Dec 01 2024</div> </div> </div> <div class=section> <div class=sectname>EXAMPLE</div> <div class=sectbody> <div class=sectline>a(6) = 4 from 6 = 4+2 = 3+3 = 2+2+2.</div> <div class=sectline>G.f. = 1 + x^2 + x^3 + 2*x^4 + 2*x^5 + 4*x^6 + 4*x^7 + 7*x^8 + 8*x^9 + ...</div> <div class=sectline>From <a href="/wiki/User:Gus_Wiseman">Gus Wiseman</a>, May 19 2019: (Start)</div> <div class=sectline>The a(2) = 1 through a(9) = 8 partitions not containing 1 are the following. The Heinz numbers of these partitions are given by <a href="/A005408" title="The odd numbers: a(n) = 2*n + 1.">A005408</a>.</div> <div class=sectline> (2) (3) (4) (5) (6) (7) (8) (9)</div> <div class=sectline> (22) (32) (33) (43) (44) (54)</div> <div class=sectline> (42) (52) (53) (63)</div> <div class=sectline> (222) (322) (62) (72)</div> <div class=sectline> (332) (333)</div> <div class=sectline> (422) (432)</div> <div class=sectline> (2222) (522)</div> <div class=sectline> (3222)</div> <div class=sectline>The a(2) = 1 through a(9) = 8 partitions of n - 1 whose least part appears exactly once are the following. The Heinz numbers of these partitions are given by <a href="/A247180" title="Numbers with nonrepeating smallest prime factor.">A247180</a>.</div> <div class=sectline> (1) (2) (3) (4) (5) (6) (7) (8)</div> <div class=sectline> (21) (31) (32) (42) (43) (53)</div> <div class=sectline> (41) (51) (52) (62)</div> <div class=sectline> (221) (321) (61) (71)</div> <div class=sectline> (331) (332)</div> <div class=sectline> (421) (431)</div> <div class=sectline> (2221) (521)</div> <div class=sectline> (3221)</div> <div class=sectline>The a(2) = 1 through a(9) = 8 partitions of n + 1 where the number of parts is itself a part are the following. The Heinz numbers of these partitions are given by <a href="/A325761" title="Heinz numbers of integer partitions whose length is itself a part.">A325761</a>.</div> <div class=sectline> (21) (22) (32) (42) (52) (62) (72) (82)</div> <div class=sectline> (311) (321) (322) (332) (333) (433)</div> <div class=sectline> (331) (431) (432) (532)</div> <div class=sectline> (4111) (4211) (531) (631)</div> <div class=sectline> (4221) (4222)</div> <div class=sectline> (4311) (4321)</div> <div class=sectline> (51111) (4411)</div> <div class=sectline> (52111)</div> <div class=sectline>The a(2) = 1 through a(8) = 7 partitions of n whose greatest part appears at least twice are the following. The Heinz numbers of these partitions are given by <a href="/A070003" title="Numbers divisible by the square of their largest prime factor.">A070003</a>.</div> <div class=sectline> (11) (111) (22) (221) (33) (331) (44)</div> <div class=sectline> (1111) (11111) (222) (2221) (332)</div> <div class=sectline> (2211) (22111) (2222)</div> <div class=sectline> (111111) (1111111) (3311)</div> <div class=sectline> (22211)</div> <div class=sectline> (221111)</div> <div class=sectline> (11111111)</div> <div class=sectline>Nonisomorphic representatives of the a(2) = 1 through a(6) = 4 2-regular multigraphs with n edges and n vertices are the following.</div> <div class=sectline> {12,12} {12,13,23} {12,12,34,34} {12,12,34,35,45} {12,12,34,34,56,56}</div> <div class=sectline> {12,13,24,34} {12,13,24,35,45} {12,12,34,35,46,56}</div> <div class=sectline> {12,13,23,45,46,56}</div> <div class=sectline> {12,13,24,35,46,56}</div> <div class=sectline>The a(2) = 1 through a(9) = 8 partitions of n with no part greater than the number of ones are the following. The Heinz numbers of these partitions are given by <a href="/A325762" title="Heinz numbers of integer partitions with no part greater than the number of ones.">A325762</a>.</div> <div class=sectline> (11) (111) (211) (2111) (2211) (22111) (22211) (33111)</div> <div class=sectline> (1111) (11111) (3111) (31111) (32111) (222111)</div> <div class=sectline> (21111) (211111) (41111) (321111)</div> <div class=sectline> (111111) (1111111) (221111) (411111)</div> <div class=sectline> (311111) (2211111)</div> <div class=sectline> (2111111) (3111111)</div> <div class=sectline> (11111111) (21111111)</div> <div class=sectline> (111111111)</div> <div class=sectline>(End)</div> </div> </div> <div class=section> <div class=sectname>MAPLE</div> <div class=sectbody> <div class=sectline>with(combstruct): ZL1:=[S, {S=Set(Cycle(Z, card>1))}, unlabeled]: seq(count(ZL1, size=n), n=0..50); # <a href="/wiki/User:Zerinvary_Lajos">Zerinvary Lajos</a>, Sep 24 2007</div> <div class=sectline>G:= {P=Set (Set (Atom, card>1))}: combstruct[gfsolve](G, unlabeled, x): seq (combstruct[count] ([P, G, unlabeled], size=i), i=0..50); # <a href="/wiki/User:Zerinvary_Lajos">Zerinvary Lajos</a>, Dec 16 2007</div> <div class=sectline>with(combstruct):a:=proc(m) [ZL, {ZL=Set(Cycle(Z, card>=m))}, unlabeled]; end: A:=a(2):seq(count(A, size=n), n=0..50); # <a href="/wiki/User:Zerinvary_Lajos">Zerinvary Lajos</a>, Jun 11 2008</div> <div class=sectline># alternative Maple program:</div> <div class=sectline><a href="/A002865" title="Number of partitions of n that do not contain 1 as a part.">A002865</a>:= proc(n) option remember; `if`(n=0, 1, add(</div> <div class=sectline> (numtheory[sigma](j)-1)*<a href="/A002865" title="Number of partitions of n that do not contain 1 as a part.">A002865</a>(n-j), j=1..n)/n)</div> <div class=sectline> end:</div> <div class=sectline>seq(<a href="/A002865" title="Number of partitions of n that do not contain 1 as a part.">A002865</a>(n), n=0..60); # <a href="/wiki/User:Alois_P._Heinz">Alois P. Heinz</a>, Sep 17 2017</div> </div> </div> <div class=section> <div class=sectname>MATHEMATICA</div> <div class=sectbody> <div class=sectline>Table[ PartitionsP[n + 1] - PartitionsP[n], {n, -1, 50}] (* <a href="/wiki/User:Robert_G._Wilson_v">Robert G. Wilson v</a>, Jul 24 2004 *)</div> <div class=sectline>f[1, 1] = 1; f[n_, k_] := f[n, k] = If[n < 0, 0, If[k > n, 0, If[k == n, 1, f[n, k + 1] + f[n - k, k]]]]; Table[ f[n, 2], {n, 50}] (* <a href="/wiki/User:Robert_G._Wilson_v">Robert G. Wilson v</a> *)</div> <div class=sectline>Table[SeriesCoefficient[Exp[Sum[x^(2*k)/(k*(1 - x^k)), {k, 1, n}]], {x, 0, n}], {n, 0, 50}] (* <a href="/wiki/User:Vaclav_Kotesovec">Vaclav Kotesovec</a>, Aug 18 2018 *)</div> <div class=sectline>CoefficientList[Series[1/QPochhammer[x^2, x], {x, 0, 50}], x] (* <a href="/wiki/User:G._C._Greubel">G. C. Greubel</a>, Nov 03 2019 *)</div> <div class=sectline>Table[Count[IntegerPartitions[n], _?(FreeQ[#, 1]&)], {n, 0, 50}] (* <a href="/wiki/User:Harvey_P._Dale">Harvey P. Dale</a>, Feb 12 2023 *)</div> </div> </div> <div class=section> <div class=sectname>PROG</div> <div class=sectbody> <div class=sectline>(PARI) {a(n) = if( n<0, 0, polcoeff( (1 - x) / eta(x + x * O(x^n)), n))};</div> <div class=sectline>(PARI) a(n)=if(n, numbpart(n)-numbpart(n-1), 1) \\ <a href="/wiki/User:Charles_R_Greathouse_IV">Charles R Greathouse IV</a>, Nov 26 2012</div> <div class=sectline>(Magma) A41 := func<n|n ge 0 select NumberOfPartitions(n) else 0>; [A41(n)-A41(n-1):n in [0..50]]; // <a href="/wiki/User:Jason_Kimberley">Jason Kimberley</a>, Jan 05 2011</div> <div class=sectline>(GAP) Concatenation([1], List([1..41], n->NrPartitions(n)-NrPartitions(n-1))); # <a href="/wiki/User:Muniru_A_Asiru">Muniru A Asiru</a>, Aug 20 2018</div> <div class=sectline>(SageMath)</div> <div class=sectline>def <a href="/A002865" title="Number of partitions of n that do not contain 1 as a part.">A002865</a>_list(prec):</div> <div class=sectline> P.<x> = PowerSeriesRing(ZZ, prec)</div> <div class=sectline> return P( 1/product((1-x^(m+2)) for m in (0..60)) ).list()</div> <div class=sectline><a href="/A002865" title="Number of partitions of n that do not contain 1 as a part.">A002865</a>_list(50) # <a href="/wiki/User:G._C._Greubel">G. C. Greubel</a>, Nov 03 2019</div> <div class=sectline>(Python)</div> <div class=sectline>from sympy import npartitions</div> <div class=sectline>def <a href="/A002865" title="Number of partitions of n that do not contain 1 as a part.">A002865</a>(n): return npartitions(n)-npartitions(n-1) if n else 1 # <a href="/wiki/User:Chai_Wah_Wu">Chai Wah Wu</a>, Mar 30 2023</div> </div> </div> <div class=section> <div class=sectname>CROSSREFS</div> <div class=sectbody> <div class=sectline>First differences of partition numbers <a href="/A000041" title="a(n) is the number of partitions of n (the partition numbers).">A000041</a>. Cf. <a href="/A053445" title="Second differences of partition numbers A000041.">A053445</a>, <a href="/A072380" title="Third differences of partition numbers A000041.">A072380</a>, <a href="/A081094" title="4th differences of partition numbers A000041.">A081094</a>, <a href="/A081095" title="5th differences of partition numbers A000041.">A081095</a>, <a href="/A232697" title="Number of partitions of 2n into parts such that the largest multiplicity equals n.">A232697</a>.</div> <div class=sectline>Pairwise sums seem to be in <a href="/A027336" title="Number of partitions of n that do not contain 2 as a part.">A027336</a>.</div> <div class=sectline>Essentially the same as <a href="/A085811" title="Number of partitions of n including 3, but not 1.">A085811</a>.</div> <div class=sectline>Cf. <a href="/A025147" title="Number of partitions of n into distinct parts >= 2.">A025147</a>, <a href="/A147768" title="Triangle read by rows: A000012^(-2) * A027293 as infinite lower triangular matrices.">A147768</a>, <a href="/A058682" title="a(n) = p(0) + p(1) + ... + p(n) - n - 1, where p = partition numbers, A000041.">A058682</a>, <a href="/A171239" title="Triangle read by rows extracted from convolution product (1,2,3,...) * A002865: (1,1,2,2,4,4,7,8,...)">A171239</a>.</div> <div class=sectline>A column of <a href="/A090824" title="Triangle read by rows: T(n,k) = number of partitions of binomial(n,k) into parts greater than k and not greater than n, 0<=k<=n.">A090824</a> and of <a href="/A133687" title="Triangle with number of equivalence classes of n X n matrices over {0,1} with rows and columns summing to k (0<=k<=n), where...">A133687</a> and of <a href="/A292508" title="Number A(n,k) of partitions of n with k kinds of 1; square array A(n,k), n>=0, k>=0, read by antidiagonals.">A292508</a> and of <a href="/A292622" title="Number A(n,k) of partitions of n with up to k distinct kinds of 1; square array A(n,k), n>=0, k>=0, read by antidiagonals.">A292622</a>. Cf. <a href="/A229161" title="Number of n X n binary matrices with exactly 2 ones in each row and column, and with rows and columns in lexicographically n...">A229161</a>.</div> <div class=sectline>2-regular not necessarily connected graphs: <a href="/A008483" title="Number of partitions of n into parts >= 3.">A008483</a> (simple graphs), <a href="/A000041" title="a(n) is the number of partitions of n (the partition numbers).">A000041</a> (multigraphs with loops allowed), this sequence (multigraphs with loops forbidden), <a href="/A027336" title="Number of partitions of n that do not contain 2 as a part.">A027336</a> (graphs with loops allowed but no multiple edges). - <a href="/wiki/User:Jason_Kimberley">Jason Kimberley</a>, Jan 05 2011</div> <div class=sectline>See also <a href="/A098743" title="Number of partitions of n into aliquant parts (i.e., parts that do not divide n).">A098743</a> (parts that do not divide n).</div> <div class=sectline>Numbers n such that in the edge-delete game on the path P_{n} the first player does not have a winning strategy: <a href="/A274161" title="Numbers n such that in the edge-delete game on the path P_{n} the first player does not have a winning strategy.">A274161</a>. - <a href="/wiki/User:Lyndsey_Wong">Lyndsey Wong</a>, Jul 09 2016</div> <div class=sectline>Cf. <a href="/A002033" title="Number of perfect partitions of n.">A002033</a>, <a href="/A070003" title="Numbers divisible by the square of their largest prime factor.">A070003</a>, <a href="/A103295" title="Number of complete rulers with length n.">A103295</a>, <a href="/A247180" title="Numbers with nonrepeating smallest prime factor.">A247180</a>, <a href="/A325676" title="Number of compositions of n such that every distinct consecutive subsequence has a different sum.">A325676</a>, <a href="/A325684" title="Number of minimal complete rulers of length n.">A325684</a>, <a href="/A325761" title="Heinz numbers of integer partitions whose length is itself a part.">A325761</a>, <a href="/A325762" title="Heinz numbers of integer partitions with no part greater than the number of ones.">A325762</a>, <a href="/A325763" title="Heinz numbers of integer partitions whose consecutive subsequence-sums cover an initial interval of positive integers.">A325763</a>.</div> <div class=sectline>Row sums of characteristic array <a href="/A145573" title="Characteristic partition array for partitions without part 1.">A145573</a>.</div> <div class=sectline>Number of partitions of n into parts >= m: <a href="/A008483" title="Number of partitions of n into parts >= 3.">A008483</a> (m = 3), <a href="/A008484" title="Number of partitions of n into parts >= 4.">A008484</a> (m = 4), <a href="/A185325" title="Number of partitions of n into parts >= 5.">A185325</a> - <a href="/A185329" title="Number of partitions of n with parts >= 9.">A185329</a> (m = 5 through 9).</div> <div class=sectline>Sequence in context: <a href="/A035989" title="Number of partitions in parts not of the form 23k, 23k+1 or 23k-1. Also number of partitions with no part of size 1 and diff...">A035989</a> <a href="/A240019" title="Number of partitions of n, where the difference between the number of odd parts and the number of even parts is 10.">A240019</a> <a href="/A036000" title="Number of partitions in parts not of the form 25k, 25k+1 or 25k-1. Also number of partitions with no part of size 1 and diff...">A036000</a> * <a href="/A085811" title="Number of partitions of n including 3, but not 1.">A085811</a> <a href="/A187219" title="Number of partitions of n that do not contain parts less than the smallest part of the partitions of n-1.">A187219</a> <a href="/A317785" title="Number of locally connected rooted trees with n nodes.">A317785</a></div> <div class=sectline>Adjacent sequences: <a href="/A002862" title="Number of nonisomorphic connected functions with no fixed points, or proper rings with n edges.">A002862</a> <a href="/A002863" title="Number of prime knots with n crossings.">A002863</a> <a href="/A002864" title="Number of alternating prime knots with n crossings.">A002864</a> * <a href="/A002866" title="a(0) = 1; for n > 0, a(n) = 2^(n-1)*n!.">A002866</a> <a href="/A002867" title="a(n) = binomial(n,floor(n/2))*(n+1)!.">A002867</a> <a href="/A002868" title="Largest number in n-th row of triangle of Lah numbers (A008297 and A271703).">A002868</a></div> </div> </div> <div class=section> <div class=sectname>KEYWORD</div> <div class=sectbody> <div class=sectline><span title="a sequence of nonnegative numbers">nonn</span>,<span title="it is very easy to produce terms of sequence">easy</span>,<span title="an exceptionally nice sequence">nice</span></div> </div> </div> <div class=section> <div class=sectname>AUTHOR</div> <div class=sectbody> <div class=sectline><a href="/wiki/User:N._J._A._Sloane">N. J. A. Sloane</a></div> </div> </div> <div class=section> <div class=sectname>STATUS</div> <div class=sectbody> <div class=sectline>approved</div> </div> </div> </div> <div class=space10></div> </div> </div></div> <p> <div class=footerpad></div> <div class=footer> <center> <div class=bottom> <div class=linksbar> <a href="/">Lookup</a> <a href="/wiki/Welcome"><font color="red">Welcome</font></a> <a href="/wiki/Main_Page"><font color="red">Wiki</font></a> <a href="/wiki/Special:RequestAccount">Register</a> <a href="/play.html">Music</a> <a href="/plot2.html">Plot 2</a> <a href="/demo1.html">Demos</a> <a href="/wiki/Index_to_OEIS">Index</a> <a href="/webcam">WebCam</a> <a href="/Submit.html">Contribute</a> <a href="/eishelp2.html">Format</a> <a href="/wiki/Style_Sheet">Style Sheet</a> <a href="/transforms.html">Transforms</a> <a href="/ol.html">Superseeker</a> <a href="/recent">Recents</a> </div> <div class=linksbar> <a href="/community.html">The OEIS Community</a> </div> <div class=linksbar> Maintained by <a href="http://oeisf.org">The OEIS Foundation Inc.</a> </div> <div class=dbinfo>Last modified February 16 08:37 EST 2025. Contains 380893 sequences.</div> <div class=legal> <a href="/wiki/Legal_Documents">License Agreements, Terms of Use, Privacy Policy</a> </div> </div> </center> </div> </body> </html>